Remove BROWSERBASE_PROJECT_ID requirement (#625)

* Remove BROWSERBASE_PROJECT_ID requirement

The Browserbase API no longer requires a project ID to create sessions —
it is inferred from the API key. Remove the env var requirement from both
the TypeScript daemon and Rust CLI, and update docs accordingly.

* Remove unnecessary Content-Type header since no body is sent

README.md

@@ -1136,7 +1136,6 @@ To enable Browserbase, use the `-p` flag:
 
 ```bash
 export BROWSERBASE_API_KEY="your-api-key"
-export BROWSERBASE_PROJECT_ID="your-project-id"
 agent-browser -p browserbase open https://example.com
 ```
 
@@ -1145,13 +1144,12 @@ Or use environment variables for CI/scripts:
 ```bash
 export AGENT_BROWSER_PROVIDER=browserbase
 export BROWSERBASE_API_KEY="your-api-key"
-export BROWSERBASE_PROJECT_ID="your-project-id"
 agent-browser open https://example.com
 ```
 
 When enabled, agent-browser connects to a Browserbase session instead of launching a local browser. All commands work identically.
 
-Get your API key and project ID from the [Browserbase Dashboard](https://browserbase.com/overview).
+Get your API key from the [Browserbase Dashboard](https://browserbase.com/overview).
 
 ### Browser Use